Expected Values Estimated via Mean-Field Approximation are 1/N-Accurate

Abstract

Mean-field approximation is a powerful tool to study large-scale stochastic systems such as data-centers -one example being the famous power of two-choice paradigm. It is shown in the literature that under quite general conditions, the empirical measure of a system of N interacting objects converges at rate O (1/ √ N ) to a deterministic dynamical system, called its mean-field approximation.In this paper, we revisit the accuracy of mean-field approximation by focusing on expected values. We show that, under almost the same general conditions, the expectation of any performance functional converges at rate O (1/N ) to its mean-field approximation. Our result applies for finite and infinite-dimensional mean-field models. We also develop a new perturbation theory argument that shows that the result holds for the stationary regime if the dynamical system is asymptotically exponentially stable. We provide numerical experiments that demonstrate that this rate of convergence is tight and that illustrate the necessity of our conditions. As an example, we apply our result to the classical two-choice model. By combining our theory with numerical experiments, we claim that, as the load ρ goes to 1, the average queue length of a two-choice system with N servers is log 2

Steady-State Analysis of the Join-the-Shortest-Queue Model in the Halfin–Whitt Regime

“…The methodology used in this paper can be discussed in terms of [16,36,40,41]. The main technical driver of our results are bounds on the derivatives of the solution to a certain first order partial differential equation (PDE) related to the fluid model of the JSQ system.…”

“…In the language of [36], we need to bound the derivatives of the DFL Lyapunov function. These derivative bounds are a standard requirement to apply Stein's method, and [16,40,41] provide sufficient conditions to bound these derivatives for a large class of PDEs. The bounds in [16,40,41] require continuity of the vector field defining the fluid model, but the JSQ fluid model does not satisfy this continuity due to a reflecting condition at the boundary.…”

“…These derivative bounds are a standard requirement to apply Stein's method, and [16,40,41] provide sufficient conditions to bound these derivatives for a large class of PDEs. The bounds in [16,40,41] require continuity of the vector field defining the fluid model, but the JSQ fluid model does not satisfy this continuity due to a reflecting condition at the boundary. To circumvent this, we leverage knowledge of how the fluid model behaves to give us an explicit expression for the PDE solution, and we bound its derivatives directly using this expression.…”

“…The bounds in (1.3) are then simply an alternative interpretation of these convergence rates. For other examples of Stein's method for fluid, or mean-field models, see [16,17,40,41]. Specifically, [40] was the first to make the connection between Stein's method and convergence rates to the mean-field equilibrium.…”

Steady-state analysis of load-balancing algorithms in the sub-Halfin–Whitt regime

“…Pioneered in [15] (and called the drift-based fluid limits method) for fluid-limit analysis and in [4, 5] for steady-state diffusion approximation, the power of Stein’s method for steady-state approximations has been recognized in a number of recent papers [3, 4, 5, 7, 8, 15, 21, 22].…”